What's a point of this?

Monday, November 19th, 2012 | Author:

I recently came across a paper using a "universal domain" to discuss "generic points" of a variety, using Weil's foundations of algebraic geometry instead of Grothendieck's. First I had to learn that stuff, then I wanted to translate it. This lead to a more systematic study of what it means to be a point of a variety or scheme, in the various different definitions.

So in this post I will explain closed points, generic points, points in general position, schematic points, generalized points, rational points, geometric points, and in particular, which of these notions can be considered a particular case of another of these. I will try to give you a hint why one wants to generalize the ordinary (closed) points of a variety that much, to answer the question in the title: "What's the point of this?".

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What is ... a vector bundle?

Thursday, November 01st, 2012 | Author:

A vector bundle is a morphism that looks locally on the target like a product of the target with a vector space.

We will call the target space the base and the space of definition the total space. The preimage of a point of the base is called the fiber.

Is that the correct mathematical definition? It doesn't mention what kind of spaces we look at, what kind of morphism I'm talking about, what the product is, locally in which sense, vector space over which field, do we allow infinite dimension, ... so it's not a mathematical definition in the pedantic sense. I will give you pedantic definitions in this article, just to satisfy my need to write down what I consider to be a good terminology.

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What is ... a reductive group?

Thursday, July 19th, 2012 | Author:

If you don't have a solid education in linear algebraic groups, you might nevertheless encounter the term "reductive groups" now and then. People keep telling you to think about GL_n, as an algebraic group, and that's a good first approximation. If you want to go a step further, some confusion can happen, since the definition of reductive groups can be given for groups over the complex numbers in two quite different ways and only one of them generalizes to reductive groups over other fields, and if one wants to do non-perfect fields, it gets even more complicated.

I want to give some very short explanations on how to think about reductive groups and semisimple algebraic groups.

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From the Langlands Correspondence for Function Fields to the Geometric Langlands Correspondence II

Tuesday, June 05th, 2012 | Author:

This post is the continuation of "From the Langlands Correspondence for Function Fields to the Geometric Langlands Correspondence I", and explains how to translate the Langlands Correspondence for function fields to a geometric question.

This post grew out of the preparation for a seminar talk on this topic and is separated in two parts, this being the second, and last part.

To repeat briefly, the Langlands Correspondence for a function field F = \mathbb{F}_q(X) of a smooth projective curve X states that certain n-dimensional irreducible l-adic Galois representations correspond (1:1) to irreducible cuspidal automorphic representations of GL_n(\mathbb{A}_F). Furthermore, the L-functions of Galois representations and automorphic representations coincide.

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From the Langlands Correspondence for Function Fields to the Geometric Langlands Correspondence I

Saturday, May 26th, 2012 | Author:

The (conjectural) Langlands correspondence for number fields gave rise to a Langlands correspondence for function fields (proved by Drinfeld and Lafforgue), where the most important difference is the absence of the infinite place which simplifies things in the latter. This, in turn, can be translated to a "geometric" Langlands correspondence for curves over fields, but there are certain differences.

First, I'm going to explain what the Langlands correspondence for function fields says, with the assumption in mind that you have been exposed to some algebra before. After that, I'm going to sketch how to go to the geometric Langlands correspondence, following Frenkel's storytelling in chapter 3 of his article "Langlands and conformal field theory". Of course, this being a blogpost, I won't repeat what Frenkel says (nor delve deeper) but try to summarise, to give an overview.

This post grew out of the preparation for a seminar talk on this topic and is separated in two parts, this being the first part. The continuation is here, discussing the geometrization.

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Properties of Scheme Morphisms

Sunday, November 06th, 2011 | Author:

To prepare for my oral exams in algebraic geometry (covering Hartshorne's book "Algebraic Geometry" Chapter II and III) I sketched an overview diagram of morphism properties in the category of noetherian schemes. Maybe this is a good cheat sheet to keep with you while reading the book for the first or second time (ok, and I dropped a "Nisnevich" for no good reason, you can ignore it).

You can get a PDF version of the image or click on it to get a readable version.

diagrammatic cheat sheet of scheme morphism properties

I'm still in the process of writing down examples and counter-examples to these properties, maybe that list will be online some day (another kind of "counterexamples in algebraic geometry").

As always, I'm happy to hear any comments (did I miss an important arrow, did I get anything wrong) -- but I should stress that the diagram works in Hartshorne-world, not in EGA-terms (this kind of confusion cost me almost one entire day trying to prove wrong statements..)

UPDATE (2011-11-18): improved diagram (more information, less colour) and higher quality PNG file.

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